ФЫ В А П Р О ВАПРОЛДЗ АПРОЛДЖЫВАПРЯЧСМИТЕЧСМИТЬБНСМИТЬБЮ ЯЧСмиDON'T USE AI | DON'T USE AI |DON'T USE AI | DON'T USE AI |DON'T USE AI |SOLVE STEP BY STEP IN DIGITAL FORMATdConsider the equation (t) + 2² [dtwith the initial condition i(0) = 0.'s(a) dz = e(t), 120,III. Find the value: i(2), wheree(t) = |1 - t\, t20ЙЦУКЕНГ ШУКЕНГШЩЕНГ ШЩЗХЪЙЦУКЕНГ ШУКI

L{i(t)}=I(s)L{ddt(i(t))}=sI(s)−i(0)L{∫0ti(t)dt}=I(s)sL{e(t)}=∫0∞e(t)e−stdt=∫01(1−t)e−stdt+∫1∞(t−1)e−…

Answered: SOLVE STEP BY STEP IN DIGITAL d…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.3: Quadratic Equations

Problem 16E

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ФЫ В А П Р О ВАПРОЛДЗ АПРОЛДЖЫВАПР
ЯЧСМИТЕЧСМИТЬБНСМИТЬБЮ ЯЧСми
DON'T USE AI | DON'T USE AI |DON'T USE AI | DON'T USE AI |DON'T USE AI |
SOLVE STEP BY STEP IN DIGITAL FORMAT
d
Consider the equation (t) + 2² [
dt
with the initial condition i(0) = 0.
's(a) dz = e(t), 120,
III. Find the value: i(2), where
e(t) = |1 - t\, t20
ЙЦУКЕНГ ШУКЕНГШЩЕНГ ШЩЗХЪЙЦУКЕНГ ШУК
I

Expert Solution

Step 1: Laplace transform

$\begin{aligned} L {i (t)} & = I (s) \\ L {\frac{d}{d t} (i (t))} & = s I (s) - i (0) \\ L {\int_{0}^{t} i (t) d t} & = \frac{I (s)}{s} \\ L {e (t)} & = \int_{0}^{\infty} e (t) e^{- s t} d t \\ = \int_{0}^{1} (1 - t) e^{- s t} d t + \int_{1}^{\infty} (t - 1) e^{- s t} d t \\ = {[- (1 - t) \frac{e^{- s t}}{s} + \frac{e^{- s t}}{s^{2}}]}_{0}^{1} + {[- (t - 1) \frac{e^{- s t}}{s} - \frac{e^{- s t}}{s^{2}}]}_{1}^{\infty} \\ = \frac{1}{s} - \frac{1}{s^{2}} + \frac{2 e^{- s}}{s^{2}} \end{aligned}$